Michael Perelman wrote:
Maybe they should have checked with the Pentagon. You all know the original purpose.
Right. There are many military applications. Just to mention one branch of game theory I'm a bit familiar with: stochastic dynamic games (differential and difference games) have been used to develop so-called "smart weaponry." The canonical game in this field was "the pursuer and the evader." At least some of the pioneers (mid 1960s) of this branch of applied math were people connected to the military (Rufus Isaacs was chief scientist at the Center for Naval Analysis, but I don't really know about L.D. Berkowitz, Avner Friedman, or W. Fleming).
But, of course, the applications of mathematical knowledge don't necessarily have to be direct in, say, management or economic policy. Usually, this kind of techniques begin to work their way in theory long before they are implemented to deal with practical estimation, management, or policy problems. In economics, mathematical programming and statistical filtering were first used in theoretical exercises. Now much more sophisticated versions of mathematical programming are embedded in software algorithms and gadgets. Even Excel can do those things and people may not be aware they are using them. So it takes a while for these methods to catch, but they have a lot of potential applicability (even in economics, in spite of Knightian uncertainty).
Differential and difference games, in particular, are an extension of optimal control a la Pontryaguin (or Bellman's dynamic programming) where instead of having one agent with a single set of controls, you have n players, each with partial control over the state of the system. Depending on how you specify the state's transition equations (or equations of motion) -- because there are obviously strategic interactions between the players -- you may wind up with a very complicated set of differential or difference equations: partial, awfully nonlinear. If you make the game stochastic, it really becomes ugly. The trick when you frame it as a game is that, instead of having a simple set of, say, Hamiltonian first-order conditions, you have n sets, and you have to reconcile them all in some kind of mutually consistent solution -- a Nash equilibrium, that is. There are a few theorems that tell you that, under certain constraints imposed on the transition equations, such solutions exist and are of the right sign. But, of course, the problem is to derive the specific solutions.
But not every problem requires that you find elegant closed-form solutions. Closed-form solutions don't even have to exist. With the current expansion in computing power, I think the whole thing looks very promising. There are tricks to simplify things (linearization, Liapunov functions, whatever works) and implement them numerically. Having some kind of simpler analytical benchmark or limit case against which to compare the complexity of reality is better than having nothing. Algorithms don't need to decide for you (if you have time to micromanage things, sure, go ahead). They just give you info you can use. So, to people who demand, "Show me the money!" I'd respond: "Why me? You show it to yourself. Learn and try."
Julio
